Question: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-z^2 - 7z + 30}{5z^3 + 25z^2 - 120z}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-1(z^2 + 7z - 30)} {5z(z^2 + 5z - 24)} $ $ a = -\dfrac{1}{5z} \cdot \dfrac{z^2 + 7z - 30}{z^2 + 5z - 24} $ Next factor the numerator and denominator. $ a = - \dfrac{1}{5z} \cdot \dfrac{(z - 3)(z + 10)}{(z - 3)(z + 8)}$ Assuming $z \neq 3$ , we can cancel the $z - 3$ $ a = - \dfrac{1}{5z} \cdot \dfrac{z + 10}{z + 8}$ Therefore: $ a = \dfrac{ -z - 10 }{ 5z(z + 8)}$, $z \neq 3$